\chapter{Introduction} \label{chap:1}
Magnetic resonance imaging (MRI), which is also called nuclear magnetic resonance imaging (NRMI) or spin imaging, is the most successful applications of the theory of nuclear magnetic resonance (NMR). MRI is also the most important development of the medical diagnostic imaging since the Wilhelm R\"{o}entgen's X-ray. Its medical applications revolutionized the clinical diagnosis. In the last several decades, six scientists were awarded Nobel Prizes in three different disciplines (physics, chemistry, and physiology or medicine) for their works related to MRI.

In an MRI experiment, the nuclear magnetization of hydrogen atoms in water in human or animal body are aligned by a powerful external magnetic field. After the aligned magnetization of the hydrogen nuclei is tipped by a radio frequency (RF) wave, a gyromagnetic field is generated. The gyromagnetic field is captured by the signal receiving coils. The received signal is interpreted as the spatial frequency response of the imaged object, so the Fourier analysis is the foundation of the image reconstruction of MRI.

The non-invasive and non-ionizing MRI is superior to computerized tomography (CT) in soft tissues imaging, such as  neurological (brain), musculoskeletal, cardiovascular, and oncological imaging. Abundant diagnostic information are provided by changing parameters of MRI system.

With the development of hardware and computing technologies, functional magnetic resonance imaging (fMRI) is becoming a standard procedure with useful applications in patients management \cite{Hennig2003}. fMRI is interested in hemodynamic responses which reflect the neural activities in the brains or spinal cords of humans or animals.

Blood Oxygenation Level Dependent (BOLD) fMRI was introduced by Ogawa \cite{Ogawa1990} and Kwong \cite{Kwong1992}. The change of neural activity in a region of the brain causes the changes in blood oxygenation. The changes, called BOLD effect, can be detected by magnetic resonance imaging (MRI). The BOLD effect is the basis for almost all fMRI experiments to map patterns of activation in the working human brain \cite{Hennig2003}.

Because the BOLD effect is reflected by some physical effects omitted in conventional MRI models, the classical reconstruction methods based on Fourier transforms must be changed to accommodate aspects of fMRI, such as long readout time and nonuniform $k$-space trajectories.

\section{Organization of the Thesis}
In this chapter, we briefly review the history and basic principles of MR imaging.

In Chapter \ref{chap:2}, we detail the major problem --- the reconstruction of single-shot parameter assessment by retrieval from signal encoding (SS-PARSE). A new iterative reconstruction methods based on a more accurate physical model is developed. After reviewing some similar works, we focus on two issues: quality improvement and fast algorithm.

In Chapter \ref{chap:3}, implement in an FFT with nonuniform sampling, in the reconstruction is investigated.

In Chapter \ref{chap:4}, we extend concepts stated in chapter 2 and 3 to parallel imaging.

Chapter \ref{chap:5} concludes the thesis. The innovative ideas are summarized and the possible future work is discussed.

\section{A Brief History of MRI}
In 1930's, Isidor Rabi investigated the relationship between nuclei, magnetic field and external RF. in 1944, his work was awarded Nobel Prize in physics ``for his resonance method for recording the magnetic properties of atomic nuclei".

In 1946, Felix Bloch \cite{Bloch1946} and Edward Purcell \cite{Purcell1946} laid physical foundation of MRI. They independently observed the phenomenon of NMR. They also theoretically explained the experiments. They were awarded Nobel Prize ``for their development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith" in 1952.

From 1950 to 1970, the major developments of NMR were in chemical and physical molecular analysis.

In 1971, Raymond Damadian \cite{Damadian1971} successfully used NMR to discriminate malign tissue from normal tissue in a rat by measuring relaxation times of the different tissues.

In 1973, nearly not published paper \cite{Lauterbur1973}, Paul Lauterbur illustrated the internal structure of a clam acquired by MRI. The door of medical application of MRI was opened. In the same decade, Peter Mansfield mathematically analyzed the RF signals of MRI and developed method for fast imaging. Lauterbur and Mansfield shared 2003 Nobel Prize in physiology or medicine ``for their discoveries concerning magnetic resonance imaging".

\section{MRI Model}
\subsection{Basic Concepts}
Spin, a quantum mechanical property, can interact with an external magnetic field $\mathbf{B}_{0}$. The nuclei, which have an odd number of protons or neutrons, have a non-zero spin or magnetic moment. Without an external magnetic field, the randomly oriented spin angular momentum cannot be detected. When an external magnetic field $B_{0}$ is applied, $M_{0}$, the magnetic moment along the external field direction, is generated \cite{Haacke1999}:
\begin{align}
    M_{0} = \frac{\rho_{0} \gamma ^{2} \hbar ^{2}}{4kT}B_{0} \label{eq:Equilibrium}
\end{align}
where $\rho_{0}$ is the spin density, $\gamma$ is a constant called gyromagnetic ratio, $\hbar\triangleq h/(2 \pi )$ in terms of Planck's quantum constant $h$, $k$ is the Boltzmann constant, and $T$ is the absolute temperature. This magnetic moment vector precesses around the external field direction with an angular frequency called the Larmor frequency given by
\begin{align}
    \omega _{0} = \gamma B_{0} \label{eq:Larmor}
\end{align}

\subsection{The 1D Imaging and the Fourier Transform}
In order to have a detectable signal, a radio-frequency (RF) magnetic field is applied for a short time to make the magnetic field that is produced by the aggregate proton spins precess along with the magnetization. This precession generates a changing flux in a nearby coil. The changing flux is a signal modulated at the Larmor frequency $\omega _{0}$. After demodulation, the received signal is given by \cite{Haacke1999}
\begin{align}
    s(t) = \int \rho (x) e^{j \phi _{G} (x,t)} dx \label{eq:1D_Imaging-Eq}
\end{align}
where $\phi _{G}(x,t) = -\gamma x \int _{0} ^{t} G(\tau) d \tau$, and the $G(t)$ is the gradient in the $z$-direction. This signal is called \textit{free induction decay} (FID).

Let
\begin{align}
    k(t) = \frac{\gamma}{2 \pi} \int _{0} ^{t} G(\tau) d\tau \label{eq:spatial_freq}
\end{align}
where $k(t)$ is called $k$-trajectory \cite{Twieg1983} which samples the spatial frequency domain. Then \eqref{eq:1D_Imaging-Eq} can be written as
\begin{align}
    s(t) = \int \rho (x) e^{-j 2 \pi k(t) x} dx \label{eq:FT-of-Spin-Density}
\end{align}
\eqref{eq:FT-of-Spin-Density} shows that signal $s(t)$ is related to the spin density of the sample $\rho(x)$ by a Fourier transform. This is the foundation of the reconstruction of conventional MRI. Two $k$-space trajectories are illustrated in Figure \ref{fig:fig-two-trajs}.
\begin{figure*}
    \centering
    \subfloat[EPI Trajectory]{\includegraphics[width=3.0in]{C:/TeX/Thesis/Chapter1/Figures/EPI_Traj.eps}}
    \subfloat[Rosette Trajectory]{\includegraphics[width=3.0in]{C:/TeX/Thesis/Chapter1/Figures/Rosette_Traj.eps}} \\
    \caption{Two MRI trajectories.}
    \label{fig:fig-two-trajs}
\end{figure*}

The concept of 1D imaging can be extended to multi-dimensional Fourier imaging.
\begin{align}
    s (t) = \int \rho \left( \mathbf{r} \right)
    e^{-j 2 \pi \mathbf{k} (t) \cdot \mathbf{r}} d \mathbf{r} \label{eq:multi-D-MRI}
\end{align}
where $\mathbf{r}$ is a multi-dimensional vector. In this case, $t$ controls the values of the vector of $\mathbf{k}(t)$, which correspond to the frequency-domain coordinates of the Fourier transform. Thus, the signal $s(t)$ can be interpreted as a sample of the Fourier transform of $\rho ( \mathbf{r} )$ at the frequency coordinate $\mathbf{k}(t)$. In this way, one-dimensional signal can sweep through an $n$-dimensional space.

\subsection{Spatial Resolution in MRI}
Limited readout time in MRI applications restricts the number of collected samples, the number of phase encodings and the coverage of $k$-space. The inversion problem based on partially covered $k$-space is called a \textit{limited-Fourier inversion} problem \cite{Haacke1999}. In the discrete case, the 1D image $\rho(x)$ is reconstructed by
\begin{align}
    \rho(x) = \Delta k \sum _{n=0}^{N-1} s(n) e^{-j 2 \pi n \Delta k x} \label{eq:Trunc-Inv}
\end{align}
where $\Delta k$ is the $k$-space sampling interval. The total width of $k$-space coverage is $W=N\Delta k$. Let $L$ be the size of the field of the view (FOV). To avoid aliasing, the Nyquist criterion must be met
\begin{align}
    \Delta k = \frac{1}{L} \le \frac{1}{A}
\end{align}
where $A$ is the physical size of the original image.

From \eqref{eq:Trunc-Inv}, we have
\begin{align}
    s(n) = \sum _{k=0}^{N-1} \rho (k) e^{-j2 \pi n k \Delta x} \label{eq:Trunc-FT}
\end{align}
$\Delta x$ is the spatial resolution, the smallest size that can be measured for a given object,
\begin{align}
    \Delta x = \frac{L}{N} = \frac{1}{N\Delta k} = \frac{1}{W} \label{eq:Spatial-Reso}
\end{align}

\section{Relaxation and Field Inhomogeneity}
In an MRI experiment, after the RF pulse is turned off, the longitudinal magnetization field begins to exponentially recover with a time constant $T_{1}$, the \textit{longitudinal relaxation time}. $T_{1}$ is also called \textit{thermal} or \textit{spin-lattice relaxation time}.

$T_{2}$ is the exponential decay rate of the FID for an ideal MR experiment. $T_{2}$ is also known as \textit{transverse relaxation}. Because of static magnetic field inhomogeneities, an observed FID decays with with an exponential constant $T_{2}^{*}$ that is smaller than $T_{2}$.

In soft tissues, the typical $T_{1}$ is about 1 second. $T_{2}$ and $T_{2}^{*}$ are at the level of milliseconds. For most biological tissues, $T_{1}$ values are typically 5 to 10 times longer than $T_{2}$ values \cite{Elster2001}.
\begin{figure}[h!]
 \centering
 \includegraphics[width=3.0in]{C:/Tex/Thesis/Chapter1/Figures/T2Decay.eps}
 \caption{Magnitude - Free induction decay (FID)}
 \label{fig:fig_T2Decay}
\end{figure}

The difference of the spin density, $T_{1}$, and $T_{2}^{*}$ among different tissues is the basis of the MRI contrast mechanism. Because the $T_{2}^{*}$ relaxation rate of blood changes depends on whether or not the hemoglobin is bound with oxygen, the $T_{2}^{*}$ values as a function of space is of interests in fMRI. The BOLD contrast is also described by $R_{2}^{*}$, the reciprocal of $T_{2}^{*}$.

The magnetic field inhomogeneity also causes phase changes in the observed signal. In Chapter \ref{chap:2}, we will model the relaxation and field inhomogeneity by revising the \eqref{eq:multi-D-MRI} \cite{Twieg2003}:
\begin{align}
    s (t) = \int \rho \left( \mathbf{r} \right)
    e^{-\left( R_{2}^{*} (\mathbf{r}) + j \omega (\mathbf{r}) \right) t}
    e^{-j 2 \pi \mathbf{k} (t) \cdot \mathbf{r}} d \mathbf{r}
\end{align}

Conventional methods to estimate $R_{2}^{*}$ and $\omega$ are all based on multi-shot MRI. \cite{Twieg2003} suggested an iterative method for single-shot MRI. We will review this method in Chapter \ref{chap:2}.

\section{functional MRI (fMRI)}
For many years, it has been known that the functions of the human brain are controlled by different areas of the cerebral cortex. Several modalities are successful in mapping brain functions onto related laminae. Positron emission tomography (PET) detects brain activities by measuring regional cerebral blood flow (rCBF). Magnetoencephalography (MEG) and electroencephalography (EEG) detect the magnetic or electronic signal generated by the activated brain, but they can hardly locate which areas the signal is from. fMRI is a noninvasive modality that efficiently maps brain function.

Before the late 1980's, the imperfections caused by $T_2^*$ was regarded as a negative factor in MRI. The technique of spin echo can be used to refocus the RF pulse and compensate $T_2^*$ relaxation. Another method is to shorten the interval between excitation RF pulse and the signal sampling.

Later, it was recognized that the paramagnetic material in blood can be used to mark blood vessels and generate effective contrast. Because deoxyhemoglobin is more paramagnetic than oxyhemoglobin (oxyhemoglobin is more diamagnetic than deoxyhemoglobin), deoxyhemoglobin is magnetically susceptible. This means that oxygen intensity change can cause a change in the MRI signal, making deoxyhemoglobin a natural contrast agent.

Because the consumption of oxygen reflects brain activities, one can map brain function by analyzing an MRI signal. The goal of fMRI is to bridge brain activation with sensory, motor and cognitive processes.

